# Transitive Set of Ordinals is Ordinal

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## Theorem

Let $x$ be a transitive set of ordinals.

Then $x$ is itself an ordinal.

## Proof

Let $x$ be a transitive set of ordinals according to the statement of the theorem.

We have from Class of All Ordinals is Well-Ordered by Subset Relation that $\On$ is well-ordered by $\subseteq$.

By Exists Ordinal Greater than Set of Ordinals there exists $\alpha$ such that $\alpha \notin x$.

Hence let $\alpha$ be the smallest ordinal not in $x$.

So all ordinals smaller than $\alpha$ are in $x$.

Hence from Ordinal equals its Initial Segment:

- $\alpha \subseteq x$

From Transitive Class of Ordinals is Subset of Ordinal not in it:

- $x \subseteq \alpha$

Hence by set equality:

- $x = \alpha$

Thus $x$ is an ordinal.

$\blacksquare$

## Sources

- 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $5$: Ordinal Numbers: $\S 2$ Ordinals and transitivity: Theorem $2.2 \ (1)$